ThreeQuadricsConical::usage="Show that if three quadrics pass through the same conic, the planes of the other three intersection conicals pass through the same line. References: Usunáriz Balanzategui, Ubaldo; Usunáriz Sala, Ignacio. Problemas de Geometría​; Wolfram Mathematica AI Assistant;Personal.";
Three quadrics pass through the same conic (in red):
In[]:=
Show[{ContourPlot3D[{x^2+y^2-1+z(x-y)==0,x^2+y^2-1+z(2x-2y+z)==0,x^2+y^2-1+z(-x+y+4z)==0},{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->Opacity[0.9],PlotPoints->60,MeshFalse],ParametricPlot3D[{Cos[θ],Sin[θ],0},{θ,0,2π},PlotStyle->{Red,Thick}]}]
Out[]=
​
The other three intersection conicals and original conical:
In[]:=
Solve[{x^2+y^2-1+z(x-y)==0,x-y==0},{x,y}]
x-
1
2
,y-
1
2
,x
1
2
,y
1
2
;
In[]:=
ShowContourPlot3D[x-y==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->Opacity[0.5]],ParametricPlot3D-
1
2
,-
1
2
,t,{t,-5,5},PlotStyle->{Black,Thick},ParametricPlot3D
1
2
,
1
2
,t,{t,-5,5},PlotStyle->{Black,Thick},ParametricPlot3D[{Cos[θ],Sin[θ],0},{θ,0,2π},PlotStyle->{Red,Thick}]
Out[]=
In[]:=
Solve[{x^2+y^2-1+z(2x-2y+z)==0,2x-2y+z==0},{x,y}]
x
1
4
-z-
8-
2
z
,y
1
4
z-
8-
2
z
,x
1
4
-z+
8-
2
z
,y
1
4
z+
8-
2
z
;
In[]:=
ShowContourPlot3D[2x-2y+z==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->Opacity[0.5]],ParametricPlot3D
1
4
-t-
8-
2
t
,
1
4
t-
8-
2
t
,t,t,-2
2
,2
2
,PlotStyle->{Black,Thick},ParametricPlot3D
1
4
-t+
8-
2
t
,
1
4
t+
8-
2
t
,t,t,-2
2
,2
2
,PlotStyle->{Black,Thick},ParametricPlot3D[{Cos[θ],Sin[θ],0},{θ,0,2π},PlotStyle->{Red,Thick}]
Out[]=
In[]:=
Solve[{x^2+y^2-1+z(-x+y+4z)==0,-x+y+4z==0},{x,y}]
x
1
2
4z-
2
1-8
2
z
,y
1
2
-4z-
2
1-8
2
z
,x
1
2
4z+
2
1-8
2
z
,y
1
2
-4z+
2
1-8
2
z
;
In[]:=
ShowContourPlot3D[-x+y+4z==0,{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->Opacity[0.5]],ParametricPlot3D
1
2
4t-
2
1-8
2
t
,
1
2
-4t-
2
1-8
2
t
,t,t,-
1
2
2
,
1
2
2
,PlotStyle->{Black,Thick},ParametricPlot3D
1
2
4t+
2
1-8
2
t
,
1
2
-4t+
2
1-8
2
t
,t,t,-
1
2
2
,
1
2
2
,PlotStyle->{Black,Thick},ParametricPlot3D[{Cos[θ],Sin[θ],0},{θ,0,2π},PlotStyle->{Red,Thick}]
Out[]=
The planes of the other three intersection conicals and intersection line:
In[]:=
Solve[{x-y==0==0,2x-2y+z==0,-x+y+4z==0},{x,y,z}]
{{yx,z0}};{t,t,0};
In[]:=
Show[{ContourPlot3D[{x-y==0,2x-2y+z==0,-x+y+4z==0},{x,-5,5},{y,-5,5},{z,-5,5},MeshFalse,ContourStyle->Opacity[0.5]],ParametricPlot3D[{t,t,0},{t,-5,5},PlotStyle{Black,Thick}]}]
Out[]=
The intersection line is :
This checks the original statement.